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Nano- and viscoelastic Beck’s column on elastic foundation

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Abstract

Beck’s type column on a Winkler type foundation is the subject of the present analysis. Instead of the Bernoulli–Euler model describing the rod, two generalized models will be adopted: Eringen’s nonlocal model corresponding to nano-rods and a viscoelastic model of fractional Kelvin–Voigt type. The analysis shows that for a nano-rod the Herrmann–Smith paradox holds whilst for a viscoelastic rod it does not.

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References

  1. Aranda-Ruiz J., Loya J., Fernández-Sáez J.: Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos. Struct. 94, 2990–3001 (2012)

    Article  Google Scholar 

  2. Atanackovic T.M.: Stability Theory of Elastic Rods. World Scientific, New Jersey (1997)

    MATH  Google Scholar 

  3. Atanackovic T.M., Novakovic B.N., Vrcelj Z.: Application of Pontryagin’s principle to bimodal optimization of nano rods. Int. J. Struct. Stab. Dyn. 12, 1250012-1–1250012–11 (2012)

    Article  MathSciNet  Google Scholar 

  4. Atanackovic T.M., Novakovic B.N., Vrcelj Z.: Shape optimization against buckling of micro- and nano-rods. Arch. Appl. Mech. 82, 1303–1311 (2012)

    Article  MATH  Google Scholar 

  5. Atanackovic T.M., Novakovic B.N., Vrcelj Z., Zorica D.: Rotating nanorod with clamped ends. Int. J. Struct. Stab. Dyn. 15, 1450050-1–1450050–8 (2015)

    Article  MathSciNet  Google Scholar 

  6. Atanackovic T.M., Pilipovic S., Stankovic B., Zorica D.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley-ISTE, London (2014)

    Book  Google Scholar 

  7. Atanackovic T.M., Stankovic B.: On a system of differential equations with fractional derivatives arising in rod theory. J. Phys. A Math. Gen. 37, 1241–1250 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Atanackovic T.M., Zorica D.: Stability of the rotating compressed nano-rod. Zeitschrift für Angewandte Mathematik und Mechanik 94, 499–504 (2014)

    Article  MathSciNet  Google Scholar 

  9. Beck M.: Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes. Zeitschrift für angewandte Mathematik und Physik 3, 225–228 (1952)

    Article  MATH  Google Scholar 

  10. Becker M., Hauger W., Winzen W.: Influence of internal and external damping on the stability of Beck’s column on an elastic foundation. J. Sound Vib. 54, 468–472 (1977)

    Article  Google Scholar 

  11. Bigoni D., Noselli G.: Experimental evidence of flutter and divergence instabilities induced by dry friction. J. Mech. Phys. Solids 59, 2208–2226 (2011)

    Article  Google Scholar 

  12. Bolotin V.V., Zhinzher N.I.: Effects of damping on stability of elastic systems subjected to nonconservative forces. Int. J. Solids Struct. 5, 965–989 (1969)

    Article  MATH  Google Scholar 

  13. Challamel N., Wang C.M.: On lateral-torsional buckling of non-local beams. Adv. Appl. Math. Mech. 2, 389–398 (2010)

    MathSciNet  Google Scholar 

  14. Elishakoff I.: Controversy associated with the so-called “follower forces”: critical overview. Appl. Mech. Rev. 58, 117–142 (2005)

    Article  Google Scholar 

  15. Elishakoff I., Pentaras D., Dujat K., Versaci C., Muscolino G., Storch J., Bucas S., Challamel N., Natsuki T., Zhang Y.Y., Wang C.M., Ghyselinck G.: Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact. ISTE and John Wiley & Sons, London, New York (2012)

    Book  Google Scholar 

  16. Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  17. Glavardanov V.B., Spasic D.T., Atanackovic T.M.: Stability and optimal shape of Pflüger micro/nano beam. Int. J. Solids Struct. 49, 2559–2567 (2012)

    Article  Google Scholar 

  18. Katsikadelis J.T., Tsiatas G.C.: Non-linear dynamic stability of damped Beck’s column with variable cross-section. Int. J. Non-Linear Mech. 42, 164–171 (2007)

    Article  Google Scholar 

  19. Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V., Amsterdam (2006)

    MATH  Google Scholar 

  20. Kirillov O.N., Seyranian A.P.: Solution to the Herrmann–Smith problem. Dokl. Phys. 47, 767–771 (2002)

    Article  Google Scholar 

  21. Koiter W.T.: Unrealistic follower forces. J. Sound Vib. 194, 636–638 (1996)

    Article  Google Scholar 

  22. Langthjem M.A., Sugiyama Y.: Dynamic stability of columns subjected to follower loads: a survey. J. Sound Vib. 238, 809–851 (2000)

    Article  Google Scholar 

  23. Li C., Lim C.W., Yu J.L., Zeng Q.C.: Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11, 257–271 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ma H., Butcher E.A.: Stability of elastic columns with periodic retarded follower forces. J. Sound Vib. 286, 849–867 (2005)

    Article  Google Scholar 

  25. MacEwen K.W.: The follower load problem and bifurcation to periodic motions. Int. J. Non-Linear Mech. 39, 555–568 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Morgan M.R., Sinha S.C.: Influence of a viscoelastic foundation on the stability of Beck’s column: an exact analysis. J. Sound Vib. 91, 85–101 (1983)

    Article  MATH  Google Scholar 

  27. Panovko J.G., Gubanova I.I.: Stability and Oscillation of Elastic Systems: Modern Concepts, Paradoxes and Errors (in Russian). Nauka, Moscow (1987)

    Google Scholar 

  28. Reddy J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    Article  MATH  Google Scholar 

  29. Ruta G.C., Elishakoff I.: Towards the resolution of the Smith-Herrmann paradox. Acta Mech. 173, 89–105 (2004)

    Article  MATH  Google Scholar 

  30. Singh A., Mukherjee R., Turner K., Shaw S.: MEMS implementation of axial and follower end forces. J. Sound Vib. 286, 637–644 (2005)

    Article  Google Scholar 

  31. Smith T.E., Herrmann G.: Stability of a beam on an elastic foundation subjected to a follower force. J. Appl. Mech. Trans. ASME. 39, 628–629 (1972)

    Article  Google Scholar 

  32. Stankovic B., Atanackovic T.M.: On a model of a viscoelastic rod. Fract. Calc. Appl. Anal. 4, 501–522 (2001)

    MATH  MathSciNet  Google Scholar 

  33. Stankovic B., Atanackovic T.M.: On a viscoelastic rod with constitutive equation containing fractional derivative of two different orders. Math. Mech. Solids 9, 629–656 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Thai H.T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)

    Article  MathSciNet  Google Scholar 

  35. Wang C.M., Xiang Y., Kitipornchai S.: Postbuckling of micro and nano rods/tubes based on nonlocal beam theory. Int. J. Appl. Mech. 1, 259–266 (2009)

    Article  Google Scholar 

  36. Wang C.M., Zhang Y.Y., He X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401–105409 (2007)

    Article  Google Scholar 

  37. Wang C.M., Zhang Y.Y., Ramesh S.S., Kitipornchai S.: Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D Appl. Phys. 39, 3904–3909 (2006)

    Article  Google Scholar 

  38. Young T.H., Juan C.S.: Dynamic stability and response of fluttered beams subjected to random follower forces. Int. J. Non-Linear Mech. 38, 889–901 (2003)

    Article  MATH  Google Scholar 

  39. Zamani V., Kharazmi E., Mukherjee R.: Asymmetric post-flutter oscillations of a cantilever due to a dynamic follower force. J. Sound Vib. 340, 253–266 (2015)

    Article  Google Scholar 

  40. Ziegler H.: Die Stabiliätskriterien der Elastomechanik. Ingenieur-Archiv 20, 49–56 (1952)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000, Novi Sad, Serbia

    Teodor M. Atanackovic

  2. College of Engineering and Science, Victoria University, Footscray Park Campus, Melbourne, VIC, Australia

    Yanni Bouras

  3. Mathematical Institute, Serbian Academy of Arts and Sciences, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Dusan Zorica

Authors
  1. Teodor M. Atanackovic
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  2. Yanni Bouras
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  3. Dusan Zorica
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Corresponding author

Correspondence to Dusan Zorica.

Additional information

This research is supported by the Serbian Ministry of Education and Science project 174005, as well as by the Secretariat for Science of Vojvodina project 114 − 451 − 1084.

Yanni Bouras acknowledges Victoria University for the financial support provided for travel expenses.

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Atanackovic, T.M., Bouras, Y. & Zorica, D. Nano- and viscoelastic Beck’s column on elastic foundation. Acta Mech 226, 2335–2345 (2015). https://doi.org/10.1007/s00707-015-1327-1

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  • DOI: https://doi.org/10.1007/s00707-015-1327-1

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